Bernouli’s theorem can be defined as the principle of energy conservation for ideal fluids in steady/streamlined flow. It describes the relationship between the pressure, velocity, and elevation in a moving fluid (liquid or gas). As per the theorem, the compressibility and viscosity (internal friction) are negligible, while the flow is steady, or laminar. It was first derived in 1738 by Swiss mathematician Daniel Bernouli. According to the theorem, the total mechanical energy of the flowing fluid comprising the energy associated with fluid pressure, the gravitational potential energy of elevation, and the kinetic energy of the fluid will remain constant. In short, according to Bernouli’s theorem, within a horizontal flow of fluid, points of higher fluid speed will have less pressure when compared to points of slower fluid speed.

Incompressible fluids will have to speed up for maintaining a constant volume flow rate when they reach a narrow constricted section. As a result, narrow nozzles on a hose will help water to speed up. If the water is speeding up at a constriction, it indicates that it is gaining kinetic energy. Giving kinetic energy implies doing work. Hence, if the portion of fluid is speeding-up, it means that something external to that portion of fluid must be doing the work for it.

Let’s look at an example, let’s see what happens when water flows along streamlines from left to right. According to (figure one), the outlined volume of water will speed up as it enters the constricted region. The force from pressure P1 on water does positive work, pushes to the right. Meanwhile the force from pressure P2 on the fluid pushes to the left as it pushes in the opposite direction to the motion of the fluid.

It should be noted that pressure on the wider/slower side P1 has to be larger than the pressure on the narrow/faster side P2. Bernouli’s principle can be defined as the inverse relationship between the pressure and speed at a point in a fluid.

Bernoulli’s Equation

Bernoulli’s equation, which is a general and mathematical form of Bernoulli’s principle, takes into consideration changes in gravitational and potential energy. Bernoulli’s equation helps us in relating the pressure, speed, and height of two points (1 & 2) in a steady streamline flowing fluid of density ‘ρ’.

To be precise, we can say that Bernouli’s theorem explores the distribution of pressure in a moving incompressible fluid.

Consider the diameter of the pipe and the temperature is constant throughout the system. (Figure 2) depicts a channel conveying a fluid from point 1 to point 2. The pump gives the energy to cause the flow in an upward direction. Let’s assume that 1 pound of fluid enters the channel at point 1. The pressure at this point will be P1 lb/ft2. The average velocity of the fluid will be V1 lb/sec and the specific volume of the fluid will be V1 ft3 /lb. Point 1 which is at height h1 will be located above the horizontal bottom plane. The potential energy of a pound of fluid at point 1 will be equal to h1 ft. lb. As the fluid is in motion with velocity V1, the kinetic energy of a pound of fluid will be equal to V12 /2gc ft. lb. It is based on the average velocity (V) of fluid in the system. The average velocity differs from the mean velocity in reality. The kinetic energy per pound of fluid flowing in the channel is derived by

Here, w is the weight of the fluid in the system. r1 indicates the radius of the channel and VL depicts the local velocity at distance r from the axis of the channel. The distribution of velocity tends to vary within the channel at different localities; we can obtain the true integral only when the actual velocity distribution is known. In every case, a different velocity distribution curve is obtained. Hence, kinetic energy in real sense is written as V2/ αgc. Here α acts as a correction factor for variations in velocities at different locations in the channel. Viscous flow α = 1 and turbulent flow α = 2.

If one pound of fluid enters the channel, it will enter against pressure P1 lb/ft2. Hence the work will be equal to P1V1 ft. done on 1 pound of fluid, it will be added to the potential energy. The sum of all three energies will represent energy of one pound of fluid which enters the section of the channel. As per the principle of conservation of mass, when a system reaches a steady-state, and one pound fluid enters at point 1, another pound of fluid gets displaced at point 2. The energy content of fluid leaving at point 2 can be shown as

Here V2 is velocity, P2 indicates pressure, and V2∗ is the specific volume of fluid at point 2.

In case there is no gain or loss of energy in the system between points 1 and 2 it acts in accordance with the principle of conservation of energy. But it has been suggested that the energy is added by the pump. The energy is ‘W’ ft. lb/lb of fluid. Some of the energy gets converted into heat through friction and gets dissipated into the environment via radiation as the system is at a constant temperature.

‘F’ ft. lb/lb of fluid is the loss of energy due to friction. The overall energies of the system under consideration that balances between points 1 and 2 is

Incase the density of fluid ‘ρ’ is expressed as lb/ft3, then

Equation (4) turns out to be

This expression is considered on the basis of one pound of fluid entering the system; energy terms in the equation (5) are per pound mass of fluid.

Applications

Bernoulli’s equation helps us in estimating the flow rate of fluid through a pipe.

It helps us in measuring the change in velocity and pressure experienced by a fluid running from a pipe of some cross-sectional area into a pipe of a different cross-sectional area. A fluid will possess increased velocity and decreased pressure when it flows from a bigger pipe to a smaller pipe. The relationship is crucial in preventing a malfunction in water pipes as it helps in maintaining stable fluid pressure. If the pressure is too high, the pipe will explode causing damage and other problems.

If we are aware of the behavior of the fluid flow in the vicinity of the foil, Bernoulli’s theorem can be used to calculate the lift force on an airfoil.

The airspeed of an aircraft is determined using its pitot tube and static port. Bernoulli’s principle is used for regulating the airspeed indicator. It helps in displaying the indicated airspeed which is appropriate to the dynamic pressure.

By converting pressure energy produced by combustion of propellants into velocity, A De Laval nozzle uses Bernoulli’s principle for creating a force.

A venturi meter or an orifice plate helps in measuring the flow speed of a fluid. It can be placed into a pipeline for reducing the diameter of the flow. In the case of an incompressible fluid, the reduction in diameter may result in an increase in the fluid flow speed. Bernoulli’s principle suggests that there must be a decrease in the pressure in the reduced diameter region.

The maximum possible drain rate for a tank with a hole/tap at the base is found to be proportional to the square root of the height of the fluid in the tank. The drain rate can be calculated using Bernoulli’s equation. Viscosity helps in lowering the drain rate. That gets reflected in the discharge coefficient – a function of the Reynolds number and the shape of the orifice.

The Bernoulli grip relies on this principle for creating a non-contact adhesive force between a surface and the gripper.